Proving chaos for a system of coupled logistic maps: A topological approach

Abstract

In the work, we prove the presence of chaotic dynamics, for suitable values of the model parameters, for the discrete-time system, composed of two coupled logistic maps, as formulated in Yousefi et al. [Discrete Dyn. Nat. Soc. 5, 161–177 (2000)], which describes two interdependent economies, characterized by two competitive markets each, with a demand link between them. In particular, we rely on the SAP (Stretching Along the Paths) method, based on a stretching relation for maps defined on sets homeomorphic to the unit square and expanding the paths along one direction. Such technique is topological in nature and allows to establish the existence of a semiconjugacy between the considered dynamical system and the Bernoulli shift, so that the main features related to the chaos of the latter (e.g., the positivity of the topological entropy) are transmitted to the former. In more detail, we apply the SAP method both to the first and to the second iterate of the map associated with our system, and we show how dealing with the second iterate, although being more demanding in terms of computations, allows for a larger freedom in the sign and in the variation range of the linking parameters for which chaos emerges. Moreover, the latter choice guarantees a good agreement with the numerical simulations, which highlight the presence of a chaotic attractor under the conditions derived for the applicability of the SAP technique to the second iterate of the map.